Question: Alicia has invented a new app that two companies are interested in purchasing for a 2-year contract. Company $\text{A}$ is offering $\$10{,}000$ for the first month and will increase the amount each following month by $\$5000$. Company $\text{B}$ is offering $\$500$ for the first month and will double their payment each following month. For which monthly payment will Company $\text{B}$ 's payment first exceed Company $\text{A}$ 's payment?
Solution: Notice that Company $\text{A}$ 's monthly payments grow linearly while Company $\text{B}$ 's payments grow exponentially. This means Company $\text{B}$ 's payments are bound to exceed Company $\text{A}$ 's payments at some point. Let's start calculating the monthly payments of each company to see when that happens. Month Company $\text{A}$ Company $\text{B}$ (Add $\$5000$ each month.) (Double each month.) $1$ $\$10{,}000$ $\$500$ $2$ $\$15{,}000$ $\$1000$ $3$ $\$20{,}000$ $\$2000$ $4$ $\$25{,}000$ $\$4000$ $5$ $\$30{,}000$ $\$8000$ $6$ $\$35{,}000$ $\$16{,}000$ $7$ $\$40{,}000$ $\$32{,}000$ $8$ $\$45{,}000$ $\$64{,}000$ In conclusion, Company $\text{B}$ 's payment will first exceed Company $\text{A}$ 's payment on month number $8$.